Asymptotically linear solutions in h1 of the 2-d defocusing nonlinear Schrödinger and Hartree equations

Justin Holmer, Nikolaos Tzirakis

Research output: Contribution to journalArticlepeer-review


In the 2-d setting, given an H1 solution v(t) to the linear Schrödinger equation i∂tv + v = 0, we prove the existence (but not uniqueness) of an H1 solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂t u + u -|u| p-1u = 0 for nonlinear powers 2 < p < 3 and the existence of an H1 solution u(t) to the defocusing Hartree equation i∂t u + u -(|x|black star|u|2)u = 0 for interaction powers 1 < γ < 2, such that ||u(t) - v(t)|| H1 → 0 as t → + ∞. This is a partial result toward the existence of well-defined continuous wave operators H 1 → H1 for these equations. For NLS in 2-d, such wave operators are known to exist for p < 3, while for p ≤ 2 it is known that they cannot exist. The Hartree equation in 2-d only makes sense for 0 < γ < 2, and it was previously known that wave operators cannot exist for 0 < γ ≤ 1, while no result was previously known in the range 1 < γ < 2. Our proof in the case of NLS applies a new estimate of CollianderGrillakisTzirakis to a strategy devised by Nakanishi. For the Hartree equation, we prove a new correlation estimate following the method of CollianderGrillakisTzirakis.

Original languageEnglish (US)
Pages (from-to)117-138
Number of pages22
JournalJournal of Hyperbolic Differential Equations
Issue number1
StatePublished - Mar 2010


  • Hartree equation
  • Schrödinger equation
  • Wave operators

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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